I can think of no living actor who takes to the stage with such ease and spontaneity as Mark Rylance. He makes the theatre feel like the place that he calls home.

Watching him at full stretch, as he is in this sensational double-bill of productions first seen over the summer at Shakespeare’s Globe, I found myself recalling the poet Gerard Manley Hopkins’s rapt description of a falcon in flight: “the achieve of, the mastery of the thing”.

He is in dazzling form in both these plays, delivering virtuosic, fresh-minted performances in roles that couldn’t be more different.

His Richard III is unlike any I have seen. All good actors find dark humour in Shakespeare’s portrait of a serial killer, but there are moments in Rylance’s superbly disconcerting performance when he seems like the funniest stand-up comedian you’ve ever seen, albeit one with a horribly withered arm.

In the great opening speech, in which he announces his determination to prove a villain, he is a chuckling cheeky chappie, eyes twinkling with pleasure, taking the audience into his confidence with a glee that is infectious. At one point the audience is laughing so much that he feigns outrage at our response, in a manner that recalls Frankie Howerd, but one also remembers with a shiver that many thought Fred West was a cheerful bloke until his appalling crimes were revealed.

But Rylance’s performance deepens and darkens, as it must, with sudden moments of psychotic fury, and in a chilling scene in which he holds his wife’s hand, and wipes away her tears as he publicly dooms her to death.

In the final section he harrowingly captures a man whose mind is at war with itself, his mouth agape with horror as he finds himself utterly alone and incapable of pitying even himself.

In Twelfth Night, in contrast, Rylance offers pure comic delight. As the grieving Olivia, even the way he moves make you laugh. He seems to glide across the stage as if on castors, at times executing what looks like a nifty three-point turn. It is wonderful to see this sad but also absurd figure waking to the wonder of love, entirely unaware that the youth he falls for is actually a woman in disguise, a deception given added piquancy in this “original practices” production by the fact that, as in Richard III, all the female characters are played, superbly, by men.

And although Rylance’s dazzling high definition performances in both Richard III and Twelfth Night are what lift this double bill into the sublime, the rest of the ensemble is also exceptionally strong.

Tim Carroll’s productions go directly to the heart of both plays with great assurance and a minimum of fuss, while Jenny Tiramani’s beautiful designs, all mellow oak, sumptuous Elizabethan costumes and warm candlelight, are a delight. The intimate atmosphere and sense of shared pleasure is also helped by having some of the audience seated in onstage boxes.

Stephen Fry strikes me as a solid rather than an inspired Malvolio, comically capturing the puffed up conceit and pomposity of the man, but largely missing the character’s poignancy. Roger Lloyd Pack offers a superb double as a devious Buckingham in Richard III, and a deliciously absurd and poignant Sir Andrew Aguecheek in Twelfth Night, while Johnny Flynn, Samuel Barnett, Paul Chahidi and James Garnon all excel in the female roles.

These are lovely productions lifted to greatness by Rylance’s glorious star performances.